Optimal. Leaf size=268 \[ \frac {2 \left (a^2-b^2\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}-\frac {\left (a^2+2 a b-b^2\right ) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d}+\frac {\left (a^2+2 a b-b^2\right ) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} d}+\frac {4 a b}{5 d \cot ^{\frac {5}{2}}(c+d x)}-\frac {4 a b}{d \sqrt {\cot (c+d x)}}+\frac {2 b^2}{7 d \cot ^{\frac {7}{2}}(c+d x)} \]
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Rubi [A] time = 0.29, antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {3673, 3542, 3529, 3534, 1168, 1162, 617, 204, 1165, 628} \[ \frac {2 \left (a^2-b^2\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}-\frac {\left (a^2+2 a b-b^2\right ) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d}+\frac {\left (a^2+2 a b-b^2\right ) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} d}+\frac {4 a b}{5 d \cot ^{\frac {5}{2}}(c+d x)}-\frac {4 a b}{d \sqrt {\cot (c+d x)}}+\frac {2 b^2}{7 d \cot ^{\frac {7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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Rule 204
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1168
Rule 3529
Rule 3534
Rule 3542
Rule 3673
Rubi steps
\begin {align*} \int \frac {(a+b \tan (c+d x))^2}{\cot ^{\frac {5}{2}}(c+d x)} \, dx &=\int \frac {(b+a \cot (c+d x))^2}{\cot ^{\frac {9}{2}}(c+d x)} \, dx\\ &=\frac {2 b^2}{7 d \cot ^{\frac {7}{2}}(c+d x)}+\int \frac {2 a b+\left (a^2-b^2\right ) \cot (c+d x)}{\cot ^{\frac {7}{2}}(c+d x)} \, dx\\ &=\frac {2 b^2}{7 d \cot ^{\frac {7}{2}}(c+d x)}+\frac {4 a b}{5 d \cot ^{\frac {5}{2}}(c+d x)}+\int \frac {a^2-b^2-2 a b \cot (c+d x)}{\cot ^{\frac {5}{2}}(c+d x)} \, dx\\ &=\frac {2 b^2}{7 d \cot ^{\frac {7}{2}}(c+d x)}+\frac {4 a b}{5 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (a^2-b^2\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\int \frac {-2 a b-\left (a^2-b^2\right ) \cot (c+d x)}{\cot ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\frac {2 b^2}{7 d \cot ^{\frac {7}{2}}(c+d x)}+\frac {4 a b}{5 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (a^2-b^2\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}-\frac {4 a b}{d \sqrt {\cot (c+d x)}}+\int \frac {-a^2+b^2+2 a b \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx\\ &=\frac {2 b^2}{7 d \cot ^{\frac {7}{2}}(c+d x)}+\frac {4 a b}{5 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (a^2-b^2\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}-\frac {4 a b}{d \sqrt {\cot (c+d x)}}+\frac {2 \operatorname {Subst}\left (\int \frac {a^2-b^2-2 a b x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{d}\\ &=\frac {2 b^2}{7 d \cot ^{\frac {7}{2}}(c+d x)}+\frac {4 a b}{5 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (a^2-b^2\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}-\frac {4 a b}{d \sqrt {\cot (c+d x)}}+\frac {\left (a^2-2 a b-b^2\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{d}+\frac {\left (a^2+2 a b-b^2\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{d}\\ &=\frac {2 b^2}{7 d \cot ^{\frac {7}{2}}(c+d x)}+\frac {4 a b}{5 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (a^2-b^2\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}-\frac {4 a b}{d \sqrt {\cot (c+d x)}}+\frac {\left (a^2-2 a b-b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 d}+\frac {\left (a^2-2 a b-b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 d}-\frac {\left (a^2+2 a b-b^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 \sqrt {2} d}-\frac {\left (a^2+2 a b-b^2\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 \sqrt {2} d}\\ &=\frac {2 b^2}{7 d \cot ^{\frac {7}{2}}(c+d x)}+\frac {4 a b}{5 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (a^2-b^2\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}-\frac {4 a b}{d \sqrt {\cot (c+d x)}}-\frac {\left (a^2+2 a b-b^2\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d}+\frac {\left (a^2+2 a b-b^2\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d}+\frac {\left (a^2-2 a b-b^2\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (a^2-2 a b-b^2\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}\\ &=-\frac {\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {\left (a^2-2 a b-b^2\right ) \tan ^{-1}\left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {2 b^2}{7 d \cot ^{\frac {7}{2}}(c+d x)}+\frac {4 a b}{5 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (a^2-b^2\right )}{3 d \cot ^{\frac {3}{2}}(c+d x)}-\frac {4 a b}{d \sqrt {\cot (c+d x)}}-\frac {\left (a^2+2 a b-b^2\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d}+\frac {\left (a^2+2 a b-b^2\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d}\\ \end {align*}
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Mathematica [C] time = 0.51, size = 80, normalized size = 0.30 \[ \frac {2 \left (a \left (5 a+14 b \cot (c+d x) \, _2F_1\left (-\frac {5}{4},1;-\frac {1}{4};-\cot ^2(c+d x)\right )\right )-5 \left (a^2-b^2\right ) \, _2F_1\left (-\frac {7}{4},1;-\frac {3}{4};-\cot ^2(c+d x)\right )\right )}{35 d \cot ^{\frac {7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \tan \left (d x + c\right ) + a\right )}^{2}}{\cot \left (d x + c\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.42, size = 1877, normalized size = 7.00 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.91, size = 224, normalized size = 0.84 \[ \frac {8 \, {\left (15 \, b^{2} + \frac {42 \, a b}{\tan \left (d x + c\right )} - \frac {210 \, a b}{\tan \left (d x + c\right )^{3}} + \frac {35 \, {\left (a^{2} - b^{2}\right )}}{\tan \left (d x + c\right )^{2}}\right )} \tan \left (d x + c\right )^{\frac {7}{2}} + 210 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 210 \, \sqrt {2} {\left (a^{2} - 2 \, a b - b^{2}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 105 \, \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - 105 \, \sqrt {2} {\left (a^{2} + 2 \, a b - b^{2}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )}{420 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^2}{{\mathrm {cot}\left (c+d\,x\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \tan {\left (c + d x \right )}\right )^{2}}{\cot ^{\frac {5}{2}}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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